A square is a polygon that has specifically four sides. (This also method that a quadrilateral has precisely four vertices, and also exactly 4 angles.)

## Meaning

A square is a polygon the has precisely four sides. (This also means that a square has specifically four vertices, and exactly 4 angles.)

Discussions of 2-D shapes sometimes refer just to the border (the heat segments that kind the edges of the figure) or come the internal as well. As soon as we talk around “dissecting” a parallelogram and rearranging the parts to form a rectangle in order to determine the area that the parallelogram, we are clearly referring come the edges and the interior. The usual definitions of polygons, however, refer only to the heat segments that type the edge of the polygon. Many of the time, context will make clear what friend mean, but you should remain mindful that in some cases you might need come clarify.

You are watching: A polygon with 3 sides and 1 right angle

Elementary college curricula generally have youngsters learn the name of special subsets of quadrilateral with details features. Here we list the distinct names. Watch the articles on each kind for your definitions and also special properties.

trapezoids (A and also J room “typical” examples, however all parallelograms likewise fit the meaning of trapezoids);parallelograms (E is the “typical” example, however all rectangles and also rhombuses also fit the definition of parallelograms);rectangles (F is the “typical” example, however all squares also fit the an interpretation of rectangles);rhombuses (C and D are the “typical” examples, yet all squares additionally fit the meaning of rhombuses);squares (B), the most special of them all.

Some K-8 curricula likewise include

kites (G and, in some interpretations H).

## What’s in a word?

quadri- (four) + -lateral (side) means “four-sided.”

Compare quadri- to Spanish cuatro. Lateral way “side” (think the football, because that example).

## Classification

Just together triangles and quadrilaterals are special species of polygons, there are plenty of subclasses that quadrilaterals.

Like every polygons the have much more than 3 sides, quadrilaterals can be convex prefer these

,
,
,
or concave like these
,
.

Quadrilaterals have the right to be share by whether or not their sides, angles, diagonals, or vertices have special properties. The category schemes teach in elementary institution involve the number of pairs the parallel sides, and also the congruence that sides, and whether or no all the angles are ideal angles (all angles room congruent).

The names of many of these unique quadrilaterals are also typically component of the primary school curriculum, though tiny else around the nature of these numbers may be studied till high school. Elementary school commonly has kids learn the name of

trapezoids (A and J space “typical” examples, but all parallelograms additionally fit the an interpretation of trapezoids);parallelograms (E is the “typical” example, yet all rectangles and rhombuses also fit the meaning of parallelograms);rectangles (F is the “typical” example, however all squares additionally fit the meaning of rectangles);rhombuses (C and D are the “typical” examples, yet all squares additionally fit the meaning of rhombuses);squares (B), the many special of castle all; and also sometimeskites (G and some encompass H).

The square is also the name of the consistent quadrilateral — one in which every sides are congruent and all angles are congruent.

Though the names that are given to individual numbers does no change, the means that they are grouped may depend on the features used to type them. In the group scheme presented above, parallelograms (B, C, D, E, and also F) have actually a place of their own (the best hand column), and also even rhombuses (B, C, D) have actually a place (the bottom row), however rectangles (F and also B) aren’t distinguished from the others. In the group scheme below, rectangles (F and also B) have the right hand column to themselves, yet parallelograms are not grouped in a means that excludes A, which is no a parallelogram.

Children in primary grades often discover it difficult to entrust anything (geometrical or otherwise) simultaneously to two categories.<1> Casual language also treats shape-names as “exclusive” rather than “inclusive.” Thus, casual language treats square and also rectangle as distinct, fairly than treating square together a special kind of rectangle, as math does. Similarly, students often tend to act rectangles and also parallelograms together disjoint classes, rather than see a rectangle together a special form of parallelograms.

Given 6 toy horses and 4 toy cows, and also asked whether there are more horses or more animals, an extremely young youngsters often price “more horses” because in classifying the toys as “horses,” they, for that moment, to exclude, them together “animals,” even though, if independently asked whether horses are animals, they will say yes.

Another possible means to classify quadrilaterals is by evaluating their diagonals. This might be available for center grade student who have actually learned about perpendicular lines and bisectors.

DiagonalsPerpendicularNot PerpendicularCongruentNot Congruent
BisectorNot BisectorBisectorNot Bisector
SquareSpecial KiteRectangleIsosceles Trapezoid

## Mathematical background

### Properties

The sum of the inner angles in a square is 360°. Student who understand the analogous an outcome for triangles can convince us of this by cutting a quadrilateral right into two triangle by drawing a diagonal: each triangle consists of 180° of edge measure, so the 2 triangles save on computer 360°.

Children can likewise experiment with this idea by

coloring or labeling the corners of a quadrilateral
,cutting the corners off
, andfitting the fancy vertices together to present that they every fit snugly about a point.

Tesselation: The truth that the four vertices fit snugly approximately a solitary point enables us to arrange four copies of a quadrilateral about a point. Nevertheless of the quadrilateral one starts with, four copies of it have the right to be arranged to fit snugly around a single point.

Multiple copies of the foursome will certainly tile the plane.
Even if one starts v a concave quadrilateral like this
, one have the right to group 4 identical duplicates of castle snugly around a suggest
, and can tile the entire aircraft with lot of copies.

### Special topics (rarely in elementary or middle school)

Cyclic quadrilaterals: For some quadrilaterals, the is possible to pass a single circle with all four of its vertices. These one-of-a-kind cases, dubbed cyclic quadrilaterals, incorporate rectangles (and as such squares) and isosceles trapezoids, but likewise other forms that have actually no one-of-a-kind name of your own. Cyclic quadrilaterals have actually the special residential or commercial property that the amount of their opposite angle is a directly angle, or 180 degrees. The implication works in the various other direction, too: any type of quadrilateral who opposite angles add up come 180 levels is a cyclic quadrilateral.

When the quadrilateral and the one passing v its vertices are both shown, the quadrilateral is said to be inscribed in ~ the circle and the one is said to be circumscribed about the quadrilateral.

Parallelograms that are not additionally rectangles can not be enrolled in a circle: they room not cyclic quadrilaterals. A circle that passes through 3 of the vertices is one of two people too big to pass v the fourth (blue circle at the left) or too little to pass v the 4th (red circle in ~ the right).

In enhancement to circumscribing circles about a square (drawing circles roughly a quadrilateral, touching each vertex), it is sometimes possible to inscribe a one (draw a circle in ~ a square so that each next of the quadrilateral is tangent to the circle). This group of quadrilaterals has no unique name the its own, but includes kites, rhombuses, and also squares in addition to other square that have actually no certain name.

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Investigations: once investigating special quadrilaterals and also their properties, college student find numerous ways to identify quadrilaterals. Some abundant explorations asking students to look for special properties of angles (congruent or supplementary), sides (parallel, perpendicular, or congruent), and also diagonals (perpendicular, bisecting, or congruent).